Question: Factor the following expression: $-2$ $x^2+$ $1$ $x+$ $10$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-2)}{(10)} &=& -20 \\ {a} + {b} &=& & & {1} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-20$ and add them together. Remember, since $-20$ is negative, one of the factors must be negative. The factors that add up to ${1}$ will be your ${a}$ and ${b}$ When ${a}$ is ${5}$ and ${b}$ is ${-4}$ $ \begin{eqnarray} {ab} &=& ({5})({-4}) &=& -20 \\ {a} + {b} &=& {5} + {-4} &=& 1 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-2}x^2 +{5}x {-4}x +{10} $ Group the terms so that there is a common factor in each group: $ ({-2}x^2 +{5}x) + ({-4}x +{10}) $ Factor out the common factors: $ x(-2x + 5) + 2(-2x + 5) $ Notice how $(-2x + 5)$ has become a common factor. Factor this out to find the answer. $(-2x + 5)(x + 2)$